What point on the line y=9x+8 is closest to the origin
1 answer:
to find the point, find the location
realize that the shortest line that crosses through the origin and y=9x+8 is perpendicular to y=9x+8
perpendicular lines have slopes that multiply to get -1
y=9x+8 has a slope of 9
9*m=-1, m=-1/9
the slope of the mystery line is -1/9
since it passes through the origin, the y intercept is 0
y=(-1/9)x is the equation of the line from the point to the origin
find the intersection of this line and the original line
set them equal to each other
(-1/9)x=9x+8
multily both sides by 9
-x=81x+72
minus 81 both sides
-82x=72
divide both sides by -82
x=-36/41
find y
y=(-1/9)x
y=(-1/9)(-36/41)
y=4/41
the point is
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It depends on the value of b
Exponential functions are defined for positive values of the base [ŧex] b [/tex]. Also, they're not defined if , or at least let's say that this is a trivial case, since it is the constant function 1.
Case 0<b<1:
If the base sits between 0 and 1, the exponential function is constantly decreasing. The explanation is quite intuitive, assume for the sake of simpleness that b is rational. If it sits between 0 and 1, it means that it can be written as a fraction p/q, with p<q. So, if you give a large exponent to b, you obtain
On the other hand, if you consider negative exponents, you switch numerator and denominator and then raise to the same exponent the fraction q/p, which gets larger and larger.
So, if 0<b<1, we have
and thus 0 is a horizontal asymptote as x tends to (positive) infinity.
Case b>1:
This case is very similar, except all roles are inverted. Now you start with a fraction p/q where p>q. So, with positive, large exponents you get
And as before, negative exponents switch numerator and denominator, so the fraction becomes q/p and thus you have
So, again, 0 is a horizontal asymptote, but this time for x tending towards negative infinite.